Numerical Methods in Shape Spaces and Optimal Branching Patterns

(1)

Numerical Methods in Shape Spaces and

Optimal Branching Patterns

Dissertation

zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn

vorgelegt von

Behrend Heeren

aus Kamp-Lintfort

(2)

(3)

Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn

am Institut f¨ur Numerische Simulation

1. Gutachter: Prof. Dr. Martin Rumpf 2. Gutachter: Prof. Dr. Mario Botsch Tag der Promotion: 14.02.2017 Erscheinungsjahr: 2017

(4)

(5)

Summary

The contribution of this thesis is twofold. The main part deals with numerical methods in the con-text of shape space analysis, where the shape space at hand is considered as a Riemannian manifold. In detail, we apply and extend the time-discrete geodesic calculus (established by Rumpf and Wirth [WBRS11, RW15]) to the space of discrete shells, i.e. triangular meshes with fixed connectivity. The essential building block is a variational time-discretization of geodesic curves, which is based on a lo-cal approximation of the squared Riemannian distance on the manifold. On physilo-cal shape spaces this approximation can be derivede.g. from adissimilarity measure. The dissimilarity measure between two shell surfaces can naturally be defined as an elastic deformation energy capturing both membrane and bending distortions. Combined with a non-conforming discretization of a physically sound thin shell modelthe time-discrete geodesic calculus applied to the space of discrete shells is shown to be suitable to solve important problems in computer graphics and animation. For example, shape interpolation and extrapolation can be performed consistently within the proposed framework, since the geodesic calculus is based on a classical Riemannian structure. In a Riemannian manifold, interpolation and extrapolation can be realized by computing shortest geodesics and by applying the exponential map, respectively. To extend the existing calculus, we introduce a generalized spline functional based on the covariant derivative along a curve in shape space whose minimizers can be considered asRiemannian splines. We establish a corresponding time-discrete functional that fits perfectly into the framework of Rumpf and Wirth, and prove this discretization to be consistent. Several numerical simulations reveal that the opti-mization of the spline functional—subject to appropriate constraints—can be used to solve themultiple interpolation problem in shape space,e.g. to realize keyframe animation. To enable efficient computa-tions in the space of discrete shells we make use of an approximative scheme,i.e. we compute classical cubic splines in the space of mesh descriptors (the space of edge lengths, triangle volumes and dihedral angles) and project the solution back to the manifold.

Based on the spline functional, we further develop a simpleregression model which generalizes linear regression to nonlinear shape spaces. In detail, a conventional data term measuring the deviation of the regression curve from the input data is augmented by the spline functional, which acts as a penalty term. In the limit, the corresponding minimizer is given by a geodesic curve that fits the given data best. Practically, the penalty approach enables to control the fitting curve in a user-defined manner. Numerical examples based on real data from anatomy and botany show the capability of the model.

Finally, we apply thestatistical analysisof elastic shape spaces presented by Rumpf and Wirth [RW09b, RW11a] to the space of discrete shells. To this end, we compute a Fr´echet mean within a class of shapes bearing highly nonlinear variations and perform a principal component analysis with respect to the met-ric induced by the Hessian of an elastic shell energy. In particular, our model is shown to outperform a standard Procrustes analysis,i.e. applying the Euclidean metric to the vector of nodal positions.

The last part of this thesis deals with the optimization of microstructures arisinge.g. at austenite-marten-site interfaces in shape memory alloys. For a corresponding scalar problem, Kohn and M¨uller [KM92, KM94] proved existence of a minimizer and provided a lower and an upper bound for the optimal energy. To establish the upper bound, they studied a particular branching pattern generated by mixing two dif-ferent martensite phases. We perform a finite element simulation based on subdivision surfaces that sug-gests a topologically different class of branching patterns to represent an optimal microstructure. Based on these observations we derive a novel, low dimensional family of patterns and show—numerically and analytically—that our new branching pattern results in a significantly better upper energy bound.

(6)

(7)

Nomenclature

Some symbols are ambiguous but unique in the present context. For a functionalW =W[y1, . . . , yn], the first Gˆateaux derivative w.r.t.yi(resp. theith argument) is denoted by either∂yiWor∂iWorW,i.

Differential geometry:

M generic Riemannian manifold or shape space, finite or infinite dimensional

g, gp generic Riemannian metric (atp∈ M)

dist Riemannian distance

x:ω ⊂_Rd_{→ M} _{(local) parametrization of finite dimensional manifold} D

dt covariant derivative along a curve

g, h first and second fund. form of an embedded surface inR3_{, respectively;} either as bilinear form or as matrix representation inR2,2

S :TpM →TpM embedded shape operator; endomorphism on tangent space

s matrix rep. of shape operator in parameter domain,i.e.s=g−1_h_∈_R2,2 Geodesic calculus:

K order resp. stepsize of time-discrete quantities; stepsize is given byτ =K−1

W approximation of squared Riemannian distance,i.e. dissimilarity measure

L, LK _{(time-discrete) length functional (of order}_K₎ E, EK _{(time-discrete) path energy (of order}_K₎ F, FK _{(time-discrete) spline energy (of order}_K₎ exp,(EXPK₎ _{(time-discrete) exponential map (of order}_K₎ log,(_K1LOG) (time-discrete) logarithm (of orderK)

P,P (time-discrete) parallel transport map Physical symbols:

S_δ ⊂_R3 _{thin shell with thickness}_{δ >}₀

δ physical thickness of a thin shell

S ⊂_R3 _{mid-layer of a thin shell; embedded surface}

φ:S →_R3 _{generic deformation of a smooth shell}

G[φ] tangential distortion tensor w.r.t.φ,i.e.G[φ] =g−1_g

φ∈R2,2 (pointwise)

W generic deformation energy; dissimilarity measure

W generic energy density

η bending parameterη=δ2

λ, µ Lam´e-Navier coefficients

Discrete shells:

S discrete shell,i.e. nodal positions of a triangle mesh

N,T,E set of nodes, faces and edges, respectively, of a triangle mesh

X, T, E node, face and edge, respectively, of a triangle mesh

Φ discrete deformation,i.e. piecewise affine transformation acting onS

W discrete deformation energy resp. discrete dissimilarity measure

GT, HT piecewise-constant, discrete first and second fund. forms as matrices inR2,2

BT matrix rep. of piecewise-constant discrete shape operator;BT =G−_T1HT G[Φ]T piecewise-constant, discrete distortion tensor

N, NT, NE normal on triangle mesh; either associated with a faceT or an edgeE

θE dihedral angle associated with an edgeE

(10)

(11)

1 Introduction

Numerical methods in shape spaces

Computer animation films have become increasingly popular within the last two decades. Artists in animation studios design sophisticated characters of rising complexity and authenticity. The character models are typically represented by triangle meshes consisting of tens or even hundreds of thousands of degrees of freedoms. Moreover, animators are able to generate movements of these artificial creatures that are almost indistinguishable from natural motions. The development of flexible and effective tools supporting artists in creating such authentic animations is linked to the mathematics of shape spaces. In particular, a comprehensive understanding of both the geometry and the physics of natural deforma-tions and motion paths of complex shapes is essential for the creation of realistic models and efficient algorithms.

Figure 1.1:Morphing by means of interpolation (orange) computed between two input shapes (gray).

Thenumerical methods in shape spacesdeveloped in this thesis are designed to be useful in various ap-plications in computer graphics and animation. Typical problems in this field of activity are for instance:

• Morphing. Given two different poses of a complex shape, one aims at computing a natural and visually appealing transformation, i.e. an interpolation, between them. In Fig. 1.1 we show the morphing between two hand shapes.

• Keyframe animation. Given a sequence of character poses, so-called keyframes, one aims at com-puting a realistic deformation path meeting all of the keyframes,i.e. amultiple interpolation. As one can see in Fig. 1.2, a smooth keyframe animation is in general not realizable by morphing consecutive keyframes.

• Shape modeling. Given a certain character pose one wants to (i) identify infinitesimal variations that represent natural movements and (ii)extrapolatethese small variations to create an animation sequence of this motion. This method is referred to as ”animation without animating”,cf. Fig. 1.3.

• Detail transfer. When animating e.g. a running sequence of a character, one first designs the physical motion on a coarse scale (for example, by using shape modeling and keyframe animation). Afterwards, the fine scale details,e.g. the wrinkles on the skin, are added to a single frame of the sequence. A desirable aim is that these details are then automatically transferred to the entire sequence in a visually appealing manner.

(12)

Figure 1.2:A keyframe animation (orange) allows for a temporally smooth interpolation of a given set of keyframe poses (gray, left), in contrast to a consecutive morphing technique (green).

What is the contribution of this thesis? Although there exists a variety of established methods to solve these problems, we identify and address two main issues in existing approaches. First, sev-eral problems are frequently tackled with independent and heuristic approaches, which often disre-gard important geometric features. For example, nonlinear operations are transferred to linear opera-tions in a different space and afterwards the linear solution is projected back onto the original space [SZGP05, WDAH10, FB11]. Thereby, some geometric intuition is lost in the final projection step and the relation between the actual nonlinear solution and the projected linear solution is often hard to un-derstand. However, it might be obvious to apply existing and established methods to new problems once a geometric understanding is at hand. Second, the investigation and incorporation of relevantphysical

properties is often missing or distorted by linearization artefacts [BS08]. For example, a geometrically accurate approach is augmented by a non-physical regularization [KMP07]. Nevertheless, the existing heuristic or linearized approaches typically outperform geometrically and physically sophisticated meth-ods in terms of efficiency and computation times. Since performance is an (if notthe) important quantity in animation business, nonlinear physics-based methods are often considered as practically not feasible. One goal of this thesis is to propose a comprehensivegeometric calculusthat induces consistent tools for multiple applications in computer graphics that allow for a geometric interpretation. This leads to the notion of shape spaces that are equipped with the mathematical structure of a Riemannian manifold. Furthermore, we aim at a soundphysical modelto realize visually appealing simulation results induced by potentially large and global deformations without using non-physical or artificial terms for regular-ization. In particular, we are interested in a model that is invariant with respect to rigid body motions. Finally, the corresponding temporal and spatial discretization is supposed to allow for robust and efficient simulations. Hence, the entire modeling and discretization problem is a delicate balancing act between geometric consistency and physical soundness on the one hand and efficiency and practicability on the other hand.

Figure 1.3:Shape modeling: we determine plausible, infinitesimal variations of a neutral cactus pose (gray) and compute extrapolations to create realistic motion paths (orange).

(13)

How can we benefit from a Riemannian structure? It is due to Kendall [Ken84] that complex shapes,

e.g. curves, images or solid materials, are considered as individual elements or points in a high or even infinite dimensional space,i.e. theshape space. Initially, this space is just a collection of shapes without any mathematical structure. In particular, most shape spaces cannot be considered as linear vector spaces. Nevertheless, one is interested in performing mathematical operations on the set of shapes, for instance, for two given shapes one wants to compute a connecting path (cf. Fig. 1.1). The notion of an optimal or shortest path then induces naturally a distance measure which allowse.g. for a statistical analysis. It is a well-established ansatz to consider a given shape space as aRiemannian manifold. In a nutshell, a Riemannian manifold can be described as a collection of points that is locally equivalent to the Euclidean space, together with a so-calledRiemannian metric,i.e. an instruction how to measure local variations. On a Riemannian manifold the notion of a connecting path and hence a (locally) shortest path, a so-calledgeodesic, is intrinsically given. Thereby, a geodesic connecting two points can be considered as the solution of the interpolation problem. Similarly, one can extrapolate by extending geodesic paths via the exponential mapor transport details via the parallel transport—both are inherent concepts in Riemannian manifold theory. Hence the mathematical structure of a Riemannian manifold leads to the solution of a couple of problems relevant e.g. in computer graphics. In chapter 2 we review different approaches to shape space modeling in the literature and in chapter 3 we provide a brief summary on relevant concepts in Riemannian manifolds.

What is a suitable time-discretization? In a Riemannian setup, shortest curves,i.e. geodesics, are defined astime-continuousobjects. To introduce a temporal discretization we make use of the so-called

time-discrete geodesic calculus developed by Martin Rumpf, Benedikt Wirth and coworkers in a se-quence of papers, cf. e.g. [Wir09, WBRS11, RW13]. This discrete calculus is based on a variational time-discretization of geodesic curves and can be applied to a wide range of shape spaces. Besides the notion of time-discrete geodesics, the framework additionally provides time-discrete analogons for sev-eral basic but crucial geometric operators,e.g. for the exponential map and for parallel transport. Under certain assumptions on the underlying manifold, Rumpf and Wirth [RW15] have shown consistency and convergence of all time-discrete operators to their corresponding continuous counterparts.

From a practical point of view, the entire variational time-discretization is based on a local approximation of the squared Riemannian distance. In particular, one does not need to have an explicit notion of the Riemannian metric—although the distance is induced by the metric, one can also recover the metric from the distance. This circumstance is useful especially when dealing with physical shape spaces: here the a priori definition of a suitable Riemannian metric is not obvious whereas the derivation of a physically meaningfuldissimilarity measureis feasible. Eventually, exactly this dissimilarity measure represents the approximation of the squared Riemannian distance. In chapter 4, we provide a brief but comprehensive summary of the time-discrete geodesic calculus proposed by Rumpf and Wirth and present a consistency proof of a time-discrete covariant derivative.

Which shape space is appropiate for animation? Although characters in animation movies appear as

solidbodies they are represented typically ashollowobjects, whose ”skin” is describede.g. by triangle meshes in the computer. This is due to the fact that the physical simulation of a solid is computationally much harder than the corresponding simulation of the surface of that solid: if the solid is represented within a (regular) grid with grid sizeh, one needsO(h−3₎_{grid points to represent the interior but only}

O(h−2₎_{points to represent the surface. To obtain visually appealing animations the surface is modeled} physically as a thin shell, which is a three-dimensional material whose thickness is very small when compared to other directions. Mathematically, a shell is represented by the midlayer of the material which is typically described by a two-dimensional surface in R3. In order to deal with shells in the computer, thismidsurfaceis discretizede.g. by a polyhedral surface, hence the shape space considered here is the space of triangle meshes, also referred to as (discrete) shell space.

(14)

However, the physical modeling starts in the continuous setup, where we consider thin shells as being made of an elastic material. Intuitively, elasticity accounts for the desired property that the object is able to be deformed without being disrupted or irreversible damaged. An elastic deformation is hence characterized by being reversible in the sense that the material will return to its initial shape and size when the outer forces are removed. The elastic energy associated with the deformation of one shell into another shell will eventually represent the dissimilarity measure on the shell space—the key ingre-dient of the variational time-discretization. Although perfect elasticity is an approximation of the real world, simulations based on elastic materials look indeed very natural and are mathematically well es-tablished [Cia88, CM08]. Nevertheless, we eventually drop the pure elastic model and considerviscous

transformations. This results in a physical model that allows for the notion of paths—a concept that is axiomatically not present in elasticity theory. From a physical point of view, one can think of a viscous deformation as the concatenation of many infinitesimally small elastic deformations with subsequent stress relaxation.

As mentioned above, the surface or shell representation seems to be advantageous in terms of the num-ber of degrees of freedom. Nevertheless, the deformation of a shell is physically much more involved than the deformation of a solid. A reason for this is that the mathematical description of a shell defor-mation depends on the first and second fundamental forms of the shell’s midsurface, which are highly nonlinear geometric objects. The differences become obvious when considering discretized surfaces,

e.g. triangle meshes. Typically, the elastic deformation of a solid can be described by means of the standard linear finite elements method (FEM) as only first derivatives of the deformation are involved in the energy. However, the numerical treatment of surface bending requires second derivatives of the deformation resp. derivatives of the normal field which cannot be described by linear FEM. In particular, it is not straightforward how to describe a derivative of a normal on a triangle mesh in the first place as a polyhedral surface is naturally equipped with a piecewise constant normal field. One opportunity to resolve this problem is to make use of higher order conforming methods,e.g. one introduces additional degrees of freedom possibly without any geometric meaning that lead to large linear systems. In contrast, we consider a non-conforming method in the spirit ofDiscrete Differential Geometry, whose degrees of freedom are given by the nodal positions only. To this end, we propose a discrete shape operator on tri-angle meshes that locally depends on six vertices which leads to adiscrete bending model. Alternatively, we make use of the popularDiscrete Shellsmodel [GHDS03] which depends on two triangles,i.e. four vertices, only, and thus represents the smallest possible stencil to realize bending.

Summing up, the appropiate shape space for our purposes is the finite dimensional space of triangle meshes, the so-calledspace of discrete shells. To enable a direct comparison of two different discrete shells, we make use of the commonfixed connectivity constraint,i.e. we postulate a one-to-one corre-spondence between all nodes and all triangles. The discrete bending model is a core ingredient of a physically sounddissimilarity measureon the space of discrete shells, which penalizes variations of the first and second fundamental forms. As explained in the previous paragraph, the designation of a shape space together with the notion of a dissimilarity measure enables a direct application of the time-discrete geodesic calculus. In chapter 5 we present a detailed description as well as an empirical validation of our discrete shell space model.

| day 54 | day 69 | day 83 | day 96 | day 109

Figure 1.4:Discrete regression curve (bright) for sugar beet input shapes (dark) at 5 different days in the vegetation period.

(15)

What are relevant applications and desirable extensions? The time-discrete geodesic calculus ap-plied to the finite dimensional space of discrete shells directly offers solutions to several problems listed above: the interpolation problem between two shapes is solved by computing a time-discrete geodesic (cf. Fig. 1.1), extrapolation is performed by means of the time-discrete exponential map and detail trans-fer can be realized by the notion of a time-discrete parallel transport. In chapter 6, we show numerous numerical simulation results and present a comprehensive qualitative and quantitative evaluation. Finally, in chapter 7 we discuss three further applications and extensions of the existing calculus:

• Riemannian splines. Themultiple interpolation problem,i.e. the computation of a smooth path interpolating more than two keyframes, cannot be solved immediately by existing tools of the geodesic calculus. For example, using piecewise geodesic interpolation generates a continuous path that exhibits discontinuities in the velocity at the keyframes. In Euclidean space, cubic splines minimize the total squared second time-derivative among all curves that pass through a given set of interpolation points—this property is related to the minimization of bending energy. We build on this observation and establish how splines can be variationally defined on general shape spaces where keyframe poses act as interpolation points. To this end we introduce a generalized spline energy based on the covariant derivative of the velocity field along the curve—an object which represents a generalization of the second time-derivative. We present a corresponding time-discrete functional that fits perfectly into the time-discrete geodesic calculus, and prove this discretization to be consistent (see chapter 4). Additionally, we show that the optimization of the spline energy— subject to appropiate constraints—can be used to solve the multiple interpolation problem,e.g. to realize keyframe animation (cf. Fig. 1.2).

• Regression. Based on the (time-discrete) spline energy we develop a (time-discrete)regression model which generalizes linear regression to shape spaces. To this end, we investigate time-discrete geodesic paths in shape space which best approximate given time indexed input shapes in a least squares sense. Here, we apply the regression model to the space ofviscous fluidic objects

and present numerical simulation results based on real data from anatomy and botany. Fig. 1.4 shows a time-discrete geodesic regression path representing the growth process of sugar beet roots over a vegetation period.

• Statistical analysis. Statistical models of the shape or appearance of a class of objects are widely used in computer graphics to model the variability over the object class. Typically, the dimension of the space of observations is orders of magnitude greater than the number of samples in the training set,i.e. the underlying shape space is sampled very sparsely. In this scenario, the quality of the model depends on the validity of the structure of the manifold. We aim at performing basic statistical operations on the space of discrete shells. In this setup, the so-called Procrustes analysis, i.e. pre-registering shapes and applying the standard Euclidean metric to the vector of nodal positions, fails to capture nonlinear shell deformations appropiately. To account for this inherent nonlinearity we apply thestatistical analysisof elastic shape spaces presented by Rumpf and Wirth [RW09b, RW11a] to the space of discrete shells and compute for example a principal component analysis of a set of facial expressions.

These applications illustrate in particular that thenumerical methods in shape spacesdeveloped in this thesis are not only limited to either the space of discrete shells nor to applications in computer graphics and animation. In fact, the geodesic calculus and the corresponding extensions discussed here can be used in various areas, for instance in computer vision and pattern recognition, medical image processing or material sciences, and can be applied to different shape spaces,e.g. the space of viscous fluidic objects (cf. Fig. 1.4).

(16)

Optimizing branching patterns

Mathematical modeling has become an important tool in materials sciences since it provides a useful guide in the search for new functionalities. Important applications include the formation of microstruc-tures and phase transitionse.g. in shape memory alloys. Shape memory alloys are metals that are able to recover their original shapes by external stimuli, e.g. by a change in temperature, after permanant deformation. The reason for this effect lies in a particular behavior of the atomic lattice, namely a solid-to-solid phase transition,e.g. at some specific temperature. If the material is at a high temperature (called austenitic phase), it prefers a cubic lattice structure, whereas at a low temperature (called marten-sitic phase), there are different preferred lattices with fewer symmetries. Most interesting are, however, intermediate temperatures where both states, i.e. martensite and austenite, are present in the material and hence also transition layers between these phases occur. In practice, one often observes complex phase mixtures at these interfaces that lead to characteristic microstructures. By the Hadamard jump condition there is a correspondence between coherent martensite-martensite resp. austenite-martensite interfaces and a rank-one connection between the gradients of the (continuous) macroscopic deforma-tion. However, typical materials do not fulfill such a condition at an austenite-single-martensite interface. In physical experiments, one often observes fine twins oftwovariants of martensite, separated from a uniform region of austenite by a transition layer bearing a certain microstructure. These observations coincide with the mathematical modeling, since a coherent austenite-twinned-martensite transition is possible—in the sense of the Hadamard jump condition—if an average deformation of two variants of martensite is compatible with the austenite.

In this thesis, we focus on a simplified mathematical model describing such a coherent phase transition between a totally rigid austenite and a twinned-martensite phase. Typically, the mathematical analy-sis of phase transitions proceeds from nonlinear elasticity theory. The central goal is to determine the microstructures and their energetics by studying a suitable energy functional for the transition layers. Al-though the general situation is described most appropiately by a three-dimensional deformation acting on a three-dimensional material, there exists a well-established reduced model based on a two-dimensional domain and a scalar-valued function [KM92]. This two-dimensional model has proven to be able to capture characteristic geometric properties of the microstructures occuring at an austenite-martensite in-terface. Although many researchers have investigated this reduced model within the last two decades, a precise description of anoptimalmicrostructure,i.e. a minimizer of the mathematical model, has not been found so far. However, there are many qualitative and quantitative results one can build on (cf.

e.g. [Con00b, Con06]). Most important, Kohn and Müller [KM94] have proven the existence of such a minimizer and derived an energy scaling law in terms of certain physical parameters. These scaling laws are optimal in the sense that they provide both an upper and a lower bound on the minimal en-ergy, whereas the constants in the bounds still differ. To realize the upper bound, Kohn and Müller constructed an explicit twinned-martensite structure represented by a certain test function. In particular, their microstructurebranches,i.e. refines in a self-similar way, when it approaches the austenite phase. In chapter 8, we investigate an improvement of the upper bound which is realized by constructing explicit test functions that are energetically advantageous. Based on the insight of a numerical experiment we shall deduce a new branching pattern, that differs both geometrically and topologically from the one con-sidered by Kohn and Müller. In particular, we show numerically and analytically that this new branching pattern produces a better upper constant in the energy bounds.

Reading suggestion Chapter 2 to chapter 7 solely deal with the primary focus of this thesis,i.e. numer-ical methods in shape spaces. A comprehensive description of our investigations of optimal branching patterns is given in chapter 8 in a self-contained way.

(17)

Collaborations and publications

The contributions and main results of this thesis are the outcome of various intense collaborations and resulted in a sequence of joint publications.

The results on time-discrete geodesic calculus in the space of shells presented in chapter 6 are joint work with Martin Rumpf, Peter Schröder (Caltech), Max Wardetzky (University of Göttingen) and Benedikt Wirth (University of Münster), and have been published in [HRWW12, HRS+14]. I contributed to the development of the time-discrete geodesic calculus, in particular its application to the space of shells, and investigated the spatial discretization using tools from discrete differential geometry.

Riemannian splines and the application to the space of discrete shells—as presented in Sec. 4.3 resp. Sec. 7.1—were investigated again in joint work with M. Rumpf, P. Schr¨oder, M. Wardetzky and B. Wirth, and resulted in another publication [HRS+16]. I contributed to the mathematical modeling of the nonlin-ear model and elaborated the simplifiedLΘA-model and its efficient implementation. Rigorous aspects of time-discrete Riemannian splines are about to be submitted [HRW16], where I provided the consistency proof as presented in Sec. 4.3.

The work on time-discrete regression presented in Sec. 7.2 was initiated by a collaboration of M. Rumpf and B. Wirth with Thomas Fletcher (University of Utah). I developed a simplification of the original, fully nonlinear model and provided the resulting implementation and numerical experiments—in coop-eration with Benjamin Berkels (RWTH Aachen University). The results were published in [BFH+13]. The work concerning the statistical analysis of an elastic shell space, which is presented in Sec. 7.3, is based on a collaboration with William Smith and Chao Zhang (University of York), and has been published in [ZHRS15]. Here, my contribution was the mathematical modeling as well as the preparation of the general framework of the implementation, whereas C. Zhang assembled the corresponding code fractions and performed the quantitative analysis (not shown in this thesis).

The investigation of optimal branching patterns in chapter 8 is joint work with Patrick W. Dondl (Uni-versity of Freiburg) and M. Rumpf. The basic code structure to perform the subdivision finite element simulations has been provided by P. Dondl, whereas I focussed on the actual numerical experiments as well as on the numerical and analytical optimization of the reduced model. The results of the numerical optimization have been published in [DHR16].

Finally, I contributed to three further publications, that are not considered in this thesis:

• S. Markett, C. Montag, B. Heeren, R. Sariyska, B. Lachmann, B. Weber, and M. Reuter. Voxel-wise eigenvector centrality mapping of the human functional connectome reveals an influence of the catechol-o-methyltransferase val158met polymorphism on the default mode and somatomotor network.Brain Structure and Function, 221:2755-2765, 2016.

• S. Markett, M. Reuter, B. Heeren, B. Lachmann, B. Weber and C. Montag. Working memory ca-pacity and the functional connectome - insights from resting-state fMRI and voxelwise eigenvector centrality mapping,Brain Imaging and Behavior, 2017,accepted.

• B. Heeren, B. Wirth, S. Paulus, H. Goldbach, H. Kuhlmann and M. Rumpf. Statistical shape analy-sis of tuber roots: a methodological case study on laser scanned sugar beets,in revision.

(18)

(19)

2 Related work

2.1 Shape spaces and applications

During the past decades, the notion of shape spaces had an increasing impact on the development of new methods in computer vision, graphics and imaging, ranging from shape morphing and modeling, to shape statistics and computational anatomy. A variety of spaces of shapes has been investigated in the literature, some of them are finite-dimensional and consider polygonal curves or triangulated surfaces as shapes, but most approaches deal with infinite-dimensional spaces of shapes. In this section, we provide a summary of related work on shape spaces, with a particular focus on those spaces that can be considered as Riemannian manifolds. Eventually, we gather relevant references concerning applications in shape spaces that are also considered in this thesis, such as interpolation and extrapolation, spline curves, regression analysis and statistical models.

Shape spaces The classical treatment of shape space is due to Kendall [Ken84], where shapes are considered as k-tuples of points in Rd, which can e.g. be interpreted as discretized curves or nodes of triangulated surfaces, equipped with a quotient metric which is given by the Euclidean metric mod-ulo translation, rotation and scaling. Linear vector spaces can be considered as shape spaces as well. However, they are usually not a priori invariant with respect to translation or rotation,i.e. shape align-ment is necessary as a preprocessing step. Examples include the vector space of landmark positions [CTCG95, PMR05, SBYA05] or Lebesgue spaces [LGF00, TYW+03, DRT06]. Further classical shape spaces are induced by the Hausdorff distance or by the Gromov-Hausdorff distance. Although the for-mer one is not invariant with respect to rigid body motion, it has for instance been used to perform shape statistics [CFKM06]. In contrast, the latter one defines an isometrically invariant distance measure be-tween shapes and has been usede.g. for shape clustering [MS05] or classification [BBK08]. However, as the lack of isometries is measured globally it is difficult to examine local isometry distortions. Of particular interest for this work are shape spaces that also have the structure of a Riemannian manifold. The study of spaces of shapes from a Riemannian perspective allows to transfer many important concepts from classical geometry to these usually infinite-dimensional spaces. For example, paths, path lengths and hence shortest paths,i.e. geodesics, are generically defined. Further classical geometric quantities, such as exponential map, logarithm, covariant derivative and parallel transport, can be transferred like-wise. In the following, we will review works related to three designated groups of shape spaces, namely the space of curves, surfaces and volumetric objects (e.g. images).

For planar curves, different Riemannian metrics have been devised. In their seminal work, Michor and Mumford [MM06] examined Riemannian metrics on the manifold of closed regular curves. They showed the L2_{-metric in tangent space to be pathologic in the sense that it leads to arbitrarily short} geodesic paths1. To overcome the issue of degenerating geodesic paths they employed a curvature-weighted L2_{-metric instead (see also [MM07]). For the same reason, Mennucci}_{et al}_{. [MYS08] used} Sobolev metrics in the tangent space of planar curves, with applications in image segmentation via active

1

Later, Michor and Mumford [MM05] showed that the vanishing geodesic distance phenomenon for theL2-metric occurs also in more general shape spaces.

(20)

contours [SYM07]. Klassenet al. [KSMJ04] proposed a framework for geodesics in the space of ar-clength parameterized curves and implemented a shooting method to find them. As Riemannian metric, they used theL2_{-metric on variations of the direction or curvature functions of the curves. Schmidt}_{et al}_. [SCC06] presented an alternative variational approach for the computation of these geodesics. Srivas-tavaet al. [SJJK06, SKJJ11] utilized an elastic string representation where curves can bend and locally stretch. To this end, they assigned different weights to theL2_{-metric on stretching and bending variations} and obtained an elastic model of curves, which allowed them to define geodesics and distances between curves in a way that is invariant to transformations including reparameterization. In a sequence of pa-pers, Bruveris and co-workers inverstigated properties of the spaces of parametrized and unparametrized curves inRd, equipped with Sobolev metrics [Bru15, BMM14, BBMM14, BBM16]. For example in [BBM16], the set of Sobolev metrics was divided into three groups: theL2_{-metric, which is on the one} hand simple and reparametrization invariant, but on the other hand unsuitable for shape analysis (cf. [MM06]), the H1_{-metric, which was shown to be well suited for numerical computations and} there-fore useful in applications, and finally higher-order Sobolev metrics, whose theoretical properties make them good candidates for use in shape analysis. In particular, the space of closed plane curves equipped with a second-order Sobolev metric is geodesically complete. Hence Riemannian metrics for spaces of curves obviously benefit from a second-order Sobolev term, which can be considered as a bending term that regularizes curvature changes. This circumstance has also been exploited in physical simulations of viscous or elastic rods, whose centerline is mathematically described by a curve (cf. [BWR+08, RW15]). With regards to the space of surfaces, Kurtek et al. [?, KKDS10] studied L2_{-metrics on special} rep-resentations of parameterized surfaces. To this end, they considered geodesic paths between surfaces parametrized over the unit sphere, using local changes of the area element as a Riemannian metric. As opposed to physics-based approaches, parameterization-based metrics are not intrinsically blind to rigid body motions. Moreover, their induced distance is related to the extrinsic difference between surfaces in ambient space, so that even isometric surfaces might be far apart from one another. A combination of ex-trinsic and inex-trinsic distance was presented in [BBK09]. Bauer and co-workers [BHM11, BHM12a, BHM12b, BHM12c] generalized weighted L2_{-metrics (introduced in [MM07] for planar curves) to} higher dimensions, i.e. to the space of surfaces described by embeddings or immersions of a given manifold. They computed geodesic equations and sectional curvatures and showed in particular that these metrics overcome the degeneracy of theL2_{-metric; corresponding numerical results were shown in} [BB11].

Most relevant for this thesis is the work of Kilianet al. [KMP07] who investigated the (finite dimensional) space of triangulated surfaces. To this end, they considered geodesics between consistently triangulated meshes, with respect to a Riemannian metric measuring the stretching of triangle edges. While this met-ric is invariant to rigid body motions, the lack of a bending term leaves a non-trivial kernel of the metmet-ric tensor, including all isometric deformations of the triangular mesh. To avoid the resulting unphysical wrinkling effects, a supplementary (non physical) regularization was incorporated. Liuet al. [LSDM10] proposed a metric on a finite simplicial complex that measures resistance of an edge to stretching and compression, quantified by changes in edge length, and bending, which is associated with directional changes. In contrast to the latter examples, we make use of the regularizing effect of a bending energy and stay entirely in a physical simulation framework.

Riemannian spaces have also been considered forvolumetricshapes, where the metric imitates a phys-ical energy dissipation induced by the deformation of a shape consisting of ductile or viscous material [FJSY09, FW06, WBRS11, RW13]. Such physical approaches tend to yield intuitive paths and allow for a simple and natural time-discretization. For example, Wirthet al. [WBRS11] defined time-discrete geodesics as minimizers of a time-discrete path energy. In the context of viscous objects, the path energy consists of a sum of elastic matching energies, whose Hessian at the identity coincides with the rate of

(21)

viscous dissipation generated by the shape deformation. This viscous dissipation model can be applied to two- and three-dimensional shapes, where the former ones represent a subset of the space of images. A different approach to obtain a Riemannian structure on the space of images is using the metamor-phosis approach [MY01, TY05b, TY05a, HTY09]. This approach generalizes the flow of diffeomor-phism [DGM98] and considers the temporal change of image intensities and image intensity varia-tions (controlled by the so-called material derivative) along motion paths. Physically, the underlying metric describes the viscous dissipation in a multipolar fluid model [N ˇS91]. Recently, Berkels et al. [BER15] applied the variational time-discretization of geodesics proposed in [WBRS11] to the Rie-mannian manifold induced by the metamorphosis approach. Moreover, the notion of optimal transport [Mon81, Kan42, Kan48], in particular the formulation proposed by Benamou and Brenier [BB00], can also used to define a Riemannian structure on the space of images, that are considered as measures (cf.

e.g. [PPKC10, PFR12, SS13]).

Only for a few nontrivial Riemannian shape spaces geodesic paths can be computed in closed form (e.g. [YMSM08, SMSY11]), else the system of geodesic ODEs has to be solved using numerical time stepping schemes (e.g. [KSMJ04, BMTY05]). Alternatively, geodesic paths connecting shapes can also be approximated via the minimization of discretized path length [SCC06] or path energy functionals [FJSY09, WBRS11]. Instead of discretizing the underlying flow the variational discretization proposed by Rumpf and Wirth [WBRS11] is based on the direct minimization of a discrete path energy subject to data given at the initial and the end time. This approach turned out to be very stable and robust, and even for very small numbers of time steps one obtains qualitatively good results [RW13, MRSS15, BER15]. Building on this variational time-discretization, Rumpf and Wirth [RW15] developed a comprehensive discrete geodesic calculus on finite- and on certain infinite-dimensional shape spaces with the structure of a Hilbert manifolds, and presented a corresponding complete convergence analysis.

A main contribution of this thesis is the application of this discrete geodesic calculus to the space of (discrete) shells. In particular, we present features and applications of the geodesic calculus, such as (i) interpolation and extrapolation techniques, (ii) spline curves, (iii) a simple regression model and (iv) a statistical analysis in shape space. In the remainder of this subsection, we report on related work concerning these aspects and distinguish our contributions.

Interpolation and extrapolation Many shape interpolation schemes in computer graphics are based on the following three-step procedure. First, select a number of geometric quantities or shape descrip-tors that determine the shape (locally). Then, based on the input shapes, compute the interpolated or extrapolated quantities. Finally, reconstruct the shape or the path in shape space, respectively, that best matches the averaged quantities. The differences between various existing methods lie in the choice of the geometric quantities. Depending on whether the quantities depend linearly or nonlinearly on the vertex positions, the reconstruction is a linear or nonlinear least-squares problem.

Examples of linear reconstruction schemes were given by Sumner et al. [SZGP05] and by Lipman et al. [LSLCO05]. The method proposed by Sumneret al. [SZGP05] uses deformation gradients of the triangles as geometric quantities to be interpolated (cf. [SP04]). However, the interpolation is nonlinear since the rotational components of the deformation gradients are extracted and nonlinearly blended by taking the shortest path in the rotation group. Naturally, the blending of the rotations is done separately for each triangle which might lead to undesirable interpolation results (cf. [WDAH10] for examples and a discussion of this issue). Instead of treating all triangles individually, Lipmanet al. [LSLCO05] pre-sented a method that takes the connectivity information of the triangle mesh into account and considers transformations that connect local frames in the mesh (cf. also [KG08]). To this end, a local coordinate frame is constructed for each vertex of the mesh and so-called connection map encode the transformation between neighbouring frames. A key property of this ansatz is that it represents the local geometry of a mesh in a rotation-invariant way—in contrast to methods based on deformation gradients. Furthermore,

(22)

the reconstruction process is performed in two steps: first the connection maps are used to compute local frames, and based on the local frames, the vertex coordinates are reconstructed. This approach is well-suited to represent deformations with rotational components (i.e. twists), but has problems when dealing with deformations that include stretching.

Examples of nonlinear reconstruction schemes were given by Winkleret al. [WDAH10] and by Fr¨ohlich and Botsch [FB11]. Winkleret al. [WDAH10] used edge lengths and dihedral angles of the triangles of a surface mesh as geometric quantities to be interpolated. In order to find the mesh that best matches the interpolated edge lengths and angles they employed a rather complicated hierarchical shape matching technique. To achieve an improvement in this direction, Fr¨ohlich and Botsch [FB11] introduced a fast reconstruction scheme for the interpolation based on edge lengths and dihedral angles. Their method interpolates between simplified meshes and uses deformation transfer [BSPG06] to map the coarse inter-polated shapes to a fine mesh—we will later pick up their method in the context of spline interpolation. Besides, there are numerous interpolation techniques not depending on a reconstruction step. A classical approach in this direction was presented by Alexaet al. [ACOL00]. They proposed a morphing technique that blends the interiors of given two- or three-dimensional shapes in an as rigid as possible manner,i.e. their method is locally least-distorting. As already mentioned above, Kilianet al. [KMP07] introduced a framework of geometric modeling in the space of triangular meshes that allows for interpolation, ex-trapolation and even parallel transport. This is closely related to our approach, however, we combine the geometric modeling with a physics-based metric which was not the case in the work by Kilian et al. [KMP07]. A more physical approach was considered by Chenet al. [CPSS10] which was motivated by a number of physics-like but heuristic algorithms in geometry processing. They advocated a simple geometric model for elasticity, i.e. the distance between the differential of a deformation and the rota-tion group is penalized. Due to its geometric non-linearity, the model does not suffer from linearizarota-tion artifacts but is computationally almost as efficient as linear elasticity.

Recently, von Tycowisz et al. [vTSSH15] introduced a scheme for real-time nonlinear interpolation which exploited the fact that the set of all possible interpolated shapes is a low-dimensional object in a high-dimensional shape space. To this end they constructed a reduced optimization problem that ap-proximates its unreduced counterpart and can be solved very efficiently. We refer to the paper of von Radziewskyet al. [vRESH16] for further applications of this very efficient ansatz and in particular to the work by Brandt et al. [BvTH16], where the reduction method was used to compute approximations to time-discrete geodesics in the sense of [WBRS11].

Although we are only dealing with the interpolation of triangle meshes embedded in three-dimensional space, we finally refer to [CWKBC13, CCW16] for related work on planar deformation techniques (cf. also the review by Alexa [Ale02]).

Riemannian splines In a finite dimensional Riemannian manifold so-called Riemannian cubic poly-nomials have been introduced first by Noakeset al. [NHP89] as smooth curvesy: [0,1]→ Mthat are stationary points of the functionalF[y] = R1

0 gy D dty,˙ D dty˙

dt satisfying certain boundary conditions. Here_dtD denotes the covariant derivative along a curve, hence this representation is referred to asintrinsic

formulation. A necessary condition for optimality of the boundary value problem is given by the Euler– Lagrange equation∂yF[y] = 0which turns out to be a fourth-order differential equation [NHP89]. In

particular, Crouch and Silva Leite [CSL95] considered themultiple interpolation problemby minimizing

F[y]subject toy(tj) =yj for a set of data points (tj, yj)1≤j≤J

⊂[0,1]× M.

Trouv´e and Vialard [TV12] presented a mathematical framework to perform interpolation on time-indexed sequences of 2D or 3D shapes where they focused on the finite dimensional case of landmarks. They developed a spline interpolation method which is related to the Riemannian cubic polynomials by Noakeset al. [NHP89]. However, their approach incorporates a control variable u with respect to the Hamiltonian equations of the geodesics which can be interpreted as|u|2 ₌ _g

y(D_d_ty,˙ D_d_ty˙). Hinkle et al. [HMFJ12] introduced a family of higher-order Riemannian polynomials defined by (D

dt)

(23)

to perform polynomial regression on Riemannian manifolds. Solutions for k = 3 are referred to as cubic polynomials. The higher-order covariant differential equation is transferred into a system of first-order covariant differential equations which is solved by a numerical integrator scheme. This method is eventually applied to the shape spaces of 2D image data represented by landmark positions.

Instead of dealing with theintrinsicformulation presented above, there are several papers dealing with anextrinsicformulation,i.e. the minimization ofR1

0 ky¨k

2 _d_t _{in the ambient space. The restriction of the} curve to the manifold is then realized as a constraint. Wallner [Wal04] proved existence of minimizers in this setup for finite dimensional manifolds, and Pottmann and Hofer [PH05] showed that these min-imizers areC2_{. Additionally, Hofer and Pottmann [HP04b] provided a method for the computation of} splines on parametric surfaces, level sets, triangle meshes and point set surfaces. Algorithmically, they alternately computed minimizers in the tangent plane and projected them back to the manifold.

Besides the variational formulations there are numerous contributions dealing withsubdivision schemes

to produce smooth interpolating curves on manifolds. Subdivision schemes for curves in linear spaces are well established and mostly based on repeated computations of local (affine) averages (cf.e.g. [Dyn92, Dyn02]). By replacing the operation of affine averaging either by a geodesic average or by projecting the affine averages onto the manifold one generates a Riemannian extension. Wallner and Dyn [WD05] showed that the Riemannian extension of cubic subdivision actually producesC1_{-curves; Wallner proved}

C2_{-smoothness for a certain class of subdivision schemes in [Wal06]. More practically, Rahman}_{et al}_. [RDS+05] proposed a Deslauriers-Dubuc interpolation in the tangent space, where the mapping between tangent space and manifold is realized by means of the exponential and logarithm.

In some applications it might be useful to seek for approximating rather than for interpolating curves with respect to a given set of data points. This can be done by replacing the hard interpolation constraints by soft penalty constraints [Rei67]. Recently, Brandtet al. [BvTH16] computed approximating curves in the space of discrete shells by using the path energyR1

0 gy( ˙y,y˙) dtas regularizer. A different approach to approximating schemes are Bezi´er curves. Having the notion of geodesics at hand one can easily transfer this concept to general manifolds. In fact, the Bezi´er curves are simply generated by applying de Casteljau’s algorithm where linear interpolation is replaced by geodesic interpolation [PN07]. This was applied to the shape space of images [ERS+15] resp. to the space of discrete shells [BvTH16]. Fi-nally, Perl [Per15] introduced a Riemannian generalization of B-splines and cardinal splines and showed numerical simulations in the space of shells (represented as subdivision surfaces as in [COS00]). In this thesis, we focus on a variational formulation based on the intrinsic functional F to solve the multiple interpolation problem. To this end, we consider general, possibly infinite dimensional manifolds and derive a consistent time-discretization, which is eventually applied to the space of discrete shells.

Regression curves Time-dependent shape statistics and shape regression has already been investigated in [DFBJ07], where the regression curve is obtained via a simultaneous kernel weighted averaging in time. In the application to brain images the kernel on shape space is linked to the Sobolev metric from the group of diffeomorphisms approach [MTY02]. A variational formulation of geodesic regression was given by Fletcher [Fle11], where for given input shapesS_iat timestithe (in a least squares sense) best

ap-proximating geodesic is computed as the minimizer of the energyE[S, v] = 1 2

P

idist2(expS(tiv),Si)

over the initial shapeSof the geodesic path and its initial velocity or momentumv. Here,dist(·,·)is the Riemannian distance andexpthe exponential map. Niethammeret al. [NHV11] presented a computa-tionally efficient method in the group of diffeomorphisms shape space which is based on duality calculus in constrained optimization. Honget al. [HJS+12] proposed a generalization allowing for image meta-morphosis,i.e. a simultaneous diffeomorphic image deformation and image intensity modulation. In contrast to these approaches, we do not minimize over the initial data of geodesic shooting but directly over the shapes along a time-discrete geodesic. Eventually, this is realized by a penalty approach,i.e. we augment the data term by the intrinsic functionalFconsidered in the context of Riemannian splines.

(24)

Statistical models Statistical models of shape have been used widely in computer vision and graphics. In a 2D setting, PCA-based models such as active shape [CTCG95] or appearance models [CET01] provide a parametric representation of shape that can be used for segmentation, tracking and recognition. In a 3D setting, they are typically used for fitting to noisy or ambiguous data or for 3D reconstruction via analysis-by-synthesis [BV99, PS09, YWS+11, SHK11, SK15]. For a recent review of statistical shape modeling see [BSBW14]. When doing statistics on manifolds the analysis is performed in a manner that respects the Riemannian geometry of the manifold. This requires Riemannian notions of concepts such as distance, mean value and covariance. In this direction, Pennec [Pen06] showed how to compute these measures for a number of geometric primitives that do not form vector spaces. Fletcheret al. [FLPJ04] went further, building statistical models on such manifolds via computation of the principal geodesics of a set of data. Freifeld and Black [FB12] used this principal geodesic analysis to build nonlinear models of human body shape variation. They defined a Lie group characterizing deformations of a triangle mesh and performed statistical analysis on the resulting Riemannian manifold. However, their manifold of deformations was not physically motivated.

Our approach is mostly related to the work presented by Rumpf and Wirth [RW09a, RW09b, RW11a]. They described a statistical analysis which is based on an elastic deformation energy and applied to vol-umetric shapes represented as boundary contours of elastic objects. We adapt their statistical framework to the space of (discrete) shells by replacing the elastic deformation energy by a suitable shell energy.

2.2 Discrete bending models

There are multiple ways how to represent an embedded surface in the computer. Typically, these ap-proaches are either classified asexplicitor asimplicitrepresentations. While implicit models are given bye.g. levelset functions or phasefield models, we will focus on explicit representations given by polyhe-dral surfaces, or more precisely, by triangle meshes. Closely related is the approximation or discretization of functions on surfaces. In this section we discuss different approaches how to discretize curvature re-lated quantites on polyhedral surfaces. Usually the corresponding necessary optimality conditions lead to a fourth-order PDE.

A classical tool for the discretization of variational problems on triangle meshes is the finite element method (FEM),cf.e.g. [Cia78, Hug87, ZT00, Bra07]. This approach can be divided into three groups, namely (i) conforming, (ii) non-conforming and (iii) mixed methods. As we are dealing with a fourth-order problem, linear C0_{-conforming finite elements do not provide the necessary regularity. Instead,} one can make use of a standard C1_{-conforming finite element method which offers a direct} discretiza-tion of the fourth-order PDE. However, one introduces addidiscretiza-tional degrees of freedom—possibly with-out any geometric meaning—that lead to large linear systems [Cia78]. Alternatively, aH2_-conforming discretization can be build upon basis functions represented as so-calledsubdivision surfaces[COS00, CO01, CSA+02, GKS02], where higher order smoothness is achieved by increasing the support of the basis functions instead of introducing additional degrees of freedom1. Of course conformality is not a necessary condition. Hence one might consider a non-conforming finite element method for discretiza-tion,e.g. an adaption of theCrouzeix-Raviartelement [WBH+07] or discontinuous Galerkin methods [KMBG09]. Other non-conforming methods seek to replace continuous geometric objects by consistent discrete counterparts instead of discretizing these geometric objects in the first place. This ise.g. the spirit ofdiscrete differential geometry(DDG), which is used in many well-established approaches in computer graphics—we will also make use of this ansatz in our discretization. In mixed methods the fourth-order PDE problem is split into two second-fourth-order PDEs, which are approximated either by linear or by quadratic finite elements [Dzi88, ES10]. A group of fundamental examples is given by the discretization

1

(25)

2.2 Discrete bending models

of Willmore’s energy [Rus05, BGN07, BGN08, BR12, PPR14]. A variety of approaches discretizing general plate and shell equations build on theDiscrete Kirchhoff Triangle[BBH80, CB98, Bar13]. How-ever, these splittings are often problem dependent and introduce additional degrees of freedom.

In the remainder of this section we review related works in geometry processing and computer graphics that also deal with deformations of thin shells and curvature approximations on polyhedral surfaces,

i.e. discrete shells. Various researchers have formulated and discretized separately the membrane and bending effects of a deformation,cf.e.g. [TPBF87]. To this end, the treatment of membrane energy for triangulated meshes follows the widely studied models of elasticity in the context of FEM. In contrast, the geometrically nonlinear treatment of bending energy, which accounts for change of curvature, is more involved. As mentioned above, numerous approaches in graphics make use of tools from DDG for the treatment of bending energy. Since the leading principle of DDG is ”Discretize theory, not equations”, a consistent discrete theory is built up from scratch. In the following we consider (i) discrete analogons of curvature related quantities, and (ii) deformation models of discrete shells,i.e. polyhedral surfaces. As mentioned above, the majority of these approaches represents a non-conforming method of discretization.

Discrete curvature Taubin [Tau95] was one of the first authors in computer vision to consider an esti-mation of the curvature tensor of a surface from a polyhedral approxiesti-mation. Using the theory of normal cycles Cohen-Steiner and Morvan [CSM03] proved convergence results for integrated mean curvature as well as the integrated second fundamental form. Their discrete notion of (integrated) mean curvature reveals the so-calleddihedral angleassociated with an edge of the mesh which is defined as the angle between the two adjacent triangle normals. However, their assumptions on the mesh are very restrictive and there are no results for pointwise convergence. Wardetzky and co-workers [War06, War08, HPW06] established a convergence analysis for the discrete mean curvature vector in this integrated setup and showed that there is in general no convergence for the correspondingpointwisemean curvature vector on general meshes. Meyer et al. [MDSB02] proposed a set of discrete differential geometry operators for triangular meshes that are supposed to provide flexible tools to approximate important geometric attributes, such as normal vectors or curvatures. They started from the notion of integrated quantities which they consider as local spatial averages and derived a pointwise, discrete definition by dividing by an associated area. Hildebrandt and Polthier [HP04a] introduced a discrete vertex-based shape operator in the context of anisotropic surface filtering, which was derived from the cotangent formula [PP93]. Grinspun et al. [GGRZ06] provided a comparison of several discrete shape operators and proposed a new operator by introducing further degrees of freedom. In detail, their matrix-valued shape operator is constant on faces and is based on a normal field associated with edges. Such an edge normal is per definition perpendicular to the corresponding edge, but the angle of rotation about this edge is another degree of freedom. It was shown that this discrete shape operator passes a collection of empirical consis-tency and convergence tests even if the mesh quality is very poor. A more rigorous approach to introduce a generalized shape operator on polyhedral surfaces was performed by Hildebrandt and Polthier [HP11]. Again, they started from a weak (i.e. integrated) definition and derived a pointwise definition by testing with approximations of delta distributions. As in [HPW06] convergence was shown under the assump-tion of totally normal convergence of the polyhedral surface, whereas here also a pointwise convergence result could be established. Finally, we refer to the work by V´asaet al. [VVP+16] for a recent review on curvature estimation on meshes.

(26)

Discrete shell deformation models Terzopouloset al. [TPBF87] first introduced the concept of elas-tically deformable modelsfor geometry or surface processing. In detail, they considered a separation of an elastic energy into the sum of a membrane and bending energy measuring variations in first and sec-ond fundamental forms, respectively. The entries of the fundamental forms were discretized by using a second-order finite differences scheme on a regular and structured quad-grid. Grinspunet al. [GHDS03] introduced their nowadays well-establishedDiscrete Shellsmodel based (i) on the dihedral angle repre-sentation of integrated mean curvature [CSM03] and (ii) on the reasoning ”pointwise notion from spatial averaging” as proposed in [MDSB02]. In particular, the discrete bending energy of their model, which is supposed to measure squared differences of mean curvature, is still widely used in computer graphics, animation and cloth simuations due to its simplicity (cf. also [BMF03]). On the other hand, this energy depends on the mesh quality such that there is no convergence of this model to its continuum equivalent as shown by numerical experiments in [GGRZ06]. Wardetzkyet al. [WBH+07, BWH+06] presentend a family ofdiscrete isometric bending modelsfor triangulated surfaces which were derived from an ax-iomatic treatment of discrete Laplace operators. For flat reference configurations and isometric surface deformations it was shown that the corresponding energies arequadratic in vertex positions. Further-more, if the discrete Laplace operator is obtained by using the non-conforming Crouzeix-Raviart element their discrete isometric bending model agrees with theDiscrete Shellsbending model [GHDS03] up to second-order in the limit of small normal displacements of the plane. The work was extended forcurved

reference configurations in [GGWZ07], where the resulting bending energies were shown to be cubic

in vertex positions. Botschet al. [BPGK06] proposed a novel framework for 3D shape modeling that achieves intuitive and robust deformations by emulating physically plausible surface behavior. Here, the surface mesh is embedded in a layer of volumetric prisms, which are coupled through nonlinear, elastic forces. To deform the mesh, the prisms are rigidly transformed while minimizing an elastic energy. Besides the nonlinear approaches mentioned above, there is an abundant literature on deformation models that include a linearization step. Linear methods are attractive for several reasons,e.g. they are fast, often simple to implement and robust,i.e. the quadratic energy has a unique global minimum (if appropriate boundary conditions are chosen). Following the comprehensive survey by Botsch and Sorkine [BK08], linear deformation approaches can be classified into (i) shell-based techniques (e.g. [KCVS98, BK04]) and (ii) methods based on differential coordinates [SP04, LSLCO05, KG08]. Methods of the first kind usually linearize the nonlinear elastic energy by replacing the change of first and second fundamental forms by first and second order partial derivatives of the displacement function. To this end, these methods typically have problems with large rotational deformations, as analyzed in [BS08]. Methods of the second kind manipulate the shapee.g. via deformation gradients [SP04] or local frames [LSLCO05, KG08], and are often translation-insensitive (see also [Sor06]).

In order to accelarate simulations that are based on nonlinear bending models one might consider the space of triangular surface descriptors,i.e. edge lengths, dihedral angles and triangle areas, instead. This results in an approximation with high computational efficiency since the involved energies arequadratic

in these variables. Based on a multiscale interpolation method by Winkler et al. [WDAH10] this ap-proach was first introduced by Fr¨ohlich and Botsch [FB11]. Alternatively, there are apap-proaches in com-puter graphics [GKS02] and cloth simulation [TWS06] that build on so-calledsubdivision finite element methods—first introduced in mechanics by Ciraket al. [COS00]. Finally, we refer to Thomaszewski and Wacker [TW06] for a review on bending models in engeneering and graphics as well as to Rumpf and Wardetzky [RW14] for a survey of methods in geometry processing related to the mechanics of thin elastic surfaces.

(27)

3 Preliminaries

We give a brief summary on topics in both geometry and physics that are related to this thesis.

3.1 Background in differential geometry

In this thesis, we consider the geometric structure of possibly high-dimensional shape spaces repre-sented as manifolds as well as the geometry ofshellsrepresented as embedded surfaces. Naturally, both concepts are associated with a non-Euclidean, i.e. a Riemannian, structure and build on notions from differential geometry. To avoid confusion, we examine a careful separation between these two concepts right from the beginning. In the first part of this section, we present a survey on differential geometry and geodesic calculus onfinitedimensional manifolds based on do Carmo’s book [dC92]. These concepts will be used later to derive a consistent generalization in infinite dimensional manifolds [Kli95, Lan95], which are supposed to represent physical shape spaces. In the second part of this section, we introduce characteristic geometric objects on two-dimensional embedded surfaces, such as curvature related quan-tities, based on [B¨ar00, dC76]. These findings will later be incorporated in the mathematical modeling of thin shells.

3.1.1 Finite dimensional Riemannian manifolds

We define a differentiable manifold Mof dimensiond < ∞ in the sense of Definition 2.1 in [dC92, chap. 0],i.e. there is a family of injective mappingsxα :ωα ⊂Rd → Mwith∪αxα(ωα) =M, such

thatx−_β1◦xα is differentiable for any pairα, βwithxα(ωα)∩xβ(ωβ) 6=∅. For convenience, we will

assume in the following that there is one global parametrizationx:ω⊂_Rd_{→ M}_with_x₍_ω_{) =}_M_{. In}

particular,xis twice differentiable, injective and regular in the sense thatDxhas full rank. Furthermore we will often drop the adjective ”differentiable” when referring to adifferentiablemanifoldM.

Definition 3.1.1(Tangent space, canonical basis). The tangent spaceTpMofMatp∈ Mis defined as

TpM={γ˙(0)|γ : (−, )→ Mis a smooth curve withγ(0) =p, >0}.

Ifx:ω → Mis a parametrization withx(ξ) =pfor someξ = (ξ1, . . . , ξd)∈ω, the set(X1, . . . , Xd)

withXi =Xi(p) =Xi(ξ) =x,i(ξ) = _∂ξi∂x(ξ)is a basis ofTpM, denoted ascanonical basis.

Following Definition 2.1 in [dC92, chap. 1] we define:

Definition 3.1.2(Riemannian metric). A Riemannian metric onMis a mappingg :p 7→ gpsuch that

gp : TpM ×TpM → Ris a bilinear, symmetric and positive-definite form, which varies smoothly in

the sense thatξ7→gij(ξ) :=gx(ξ)(Xi(ξ), Xj(ξ))is a differentiable function inω. A manifold equipped

with a Riemannian

Numerical Methods in Shape Spaces and Optimal Branching Patterns